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Properties

Label 220.3.w.a.63.11
Level $220$
Weight $3$
Character 220.63
Analytic conductor $5.995$
Analytic rank $0$
Dimension $544$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [220,3,Mod(7,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 5, 14])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.w (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 63.11
Character \(\chi\) \(=\) 220.63
Dual form 220.3.w.a.7.11

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.75740 + 0.954750i) q^{2} +(2.01897 + 1.02872i) q^{3} +(2.17690 - 3.35575i) q^{4} +(4.19108 + 2.72669i) q^{5} +(-4.53030 + 0.119746i) q^{6} +(2.63142 + 5.16446i) q^{7} +(-0.621782 + 7.97580i) q^{8} +(-2.27209 - 3.12726i) q^{9} +(-9.96871 - 0.790451i) q^{10} +(9.84371 - 4.90931i) q^{11} +(7.84722 - 4.53575i) q^{12} +(-2.39288 + 15.1080i) q^{13} +(-9.55523 - 6.56366i) q^{14} +(5.65667 + 9.81653i) q^{15} +(-6.52218 - 14.6103i) q^{16} +(-1.91459 - 12.0882i) q^{17} +(6.97872 + 3.32657i) q^{18} +(24.2978 - 7.89483i) q^{19} +(18.2737 - 8.12849i) q^{20} +13.1339i q^{21} +(-12.6122 + 18.0259i) q^{22} +(-11.4273 + 11.4273i) q^{23} +(-9.46019 + 15.4633i) q^{24} +(10.1303 + 22.8556i) q^{25} +(-10.2192 - 28.8355i) q^{26} +(-4.56045 - 28.7936i) q^{27} +(23.0590 + 2.41212i) q^{28} +(-16.0454 + 49.3826i) q^{29} +(-19.3134 - 11.8509i) q^{30} +(-7.84453 - 10.7971i) q^{31} +(25.4113 + 19.4491i) q^{32} +(24.9244 + 0.214638i) q^{33} +(14.9059 + 19.4159i) q^{34} +(-3.05338 + 28.8197i) q^{35} +(-15.4404 + 0.816824i) q^{36} +(4.00077 + 7.85195i) q^{37} +(-35.1633 + 37.0727i) q^{38} +(-20.3730 + 28.0411i) q^{39} +(-24.3535 + 31.7318i) q^{40} +(-75.7259 + 24.6048i) q^{41} +(-12.5396 - 23.0814i) q^{42} +(54.5989 + 54.5989i) q^{43} +(4.95437 - 43.7202i) q^{44} +(-0.995428 - 19.3019i) q^{45} +(9.17211 - 30.9925i) q^{46} +(22.4582 - 44.0767i) q^{47} +(1.86179 - 36.2072i) q^{48} +(9.05424 - 12.4621i) q^{49} +(-39.6243 - 30.4944i) q^{50} +(8.56986 - 26.3753i) q^{51} +(45.4898 + 40.9187i) q^{52} +(-21.8861 - 3.46642i) q^{53} +(35.5052 + 46.2477i) q^{54} +(54.6420 + 6.26545i) q^{55} +(-42.8269 + 17.7765i) q^{56} +(57.1780 + 9.05611i) q^{57} +(-18.9499 - 102.104i) q^{58} +(15.0901 - 46.4424i) q^{59} +(45.2559 + 2.38726i) q^{60} +(55.4322 - 76.2959i) q^{61} +(24.0945 + 11.4852i) q^{62} +(10.1718 - 19.9633i) q^{63} +(-63.2268 - 9.91841i) q^{64} +(-51.2237 + 56.7944i) q^{65} +(-44.0071 + 23.4194i) q^{66} +(5.11791 + 5.11791i) q^{67} +(-44.7330 - 19.8900i) q^{68} +(-34.8268 + 11.3159i) q^{69} +(-22.1496 - 53.5630i) q^{70} +(-10.7606 + 14.8107i) q^{71} +(26.3552 - 16.1773i) q^{72} +(-46.5762 - 91.4110i) q^{73} +(-14.5276 - 9.97927i) q^{74} +(-3.05911 + 56.5659i) q^{75} +(26.4008 - 98.7237i) q^{76} +(51.2569 + 37.9190i) q^{77} +(9.03133 - 68.7305i) q^{78} +(-47.7369 - 65.7043i) q^{79} +(12.5028 - 79.0170i) q^{80} +(9.66242 - 29.7379i) q^{81} +(109.589 - 115.540i) q^{82} +(-9.70086 - 61.2488i) q^{83} +(44.0740 + 28.5912i) q^{84} +(24.9367 - 55.8832i) q^{85} +(-148.080 - 43.8237i) q^{86} +(-83.1957 + 83.1957i) q^{87} +(33.0350 + 81.5640i) q^{88} +11.2250i q^{89} +(20.1779 + 32.9708i) q^{90} +(-84.3215 + 27.3977i) q^{91} +(13.4711 + 63.2234i) q^{92} +(-4.73074 - 29.8687i) q^{93} +(2.61421 + 98.9023i) q^{94} +(123.361 + 33.1647i) q^{95} +(31.2970 + 65.4081i) q^{96} +(7.98092 - 50.3895i) q^{97} +(-4.01373 + 30.5454i) q^{98} +(-37.7185 - 19.6295i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 10 q^{2} - 12 q^{5} - 20 q^{6} - 10 q^{8} - 28 q^{12} - 20 q^{13} - 36 q^{16} - 20 q^{17} - 10 q^{18} - 40 q^{20} + 86 q^{22} - 12 q^{25} + 140 q^{26} - 10 q^{28} - 370 q^{30} - 100 q^{33} - 476 q^{36}+ \cdots + 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.75740 + 0.954750i −0.878700 + 0.477375i
\(3\) 2.01897 + 1.02872i 0.672990 + 0.342905i 0.756865 − 0.653571i \(-0.226730\pi\)
−0.0838758 + 0.996476i \(0.526730\pi\)
\(4\) 2.17690 − 3.35575i 0.544226 − 0.838939i
\(5\) 4.19108 + 2.72669i 0.838216 + 0.545338i
\(6\) −4.53030 + 0.119746i −0.755050 + 0.0199577i
\(7\) 2.63142 + 5.16446i 0.375918 + 0.737780i 0.999015 − 0.0443641i \(-0.0141262\pi\)
−0.623098 + 0.782144i \(0.714126\pi\)
\(8\) −0.621782 + 7.97580i −0.0777227 + 0.996975i
\(9\) −2.27209 − 3.12726i −0.252454 − 0.347474i
\(10\) −9.96871 − 0.790451i −0.996871 − 0.0790451i
\(11\) 9.84371 − 4.90931i 0.894883 − 0.446301i
\(12\) 7.84722 − 4.53575i 0.653935 − 0.377979i
\(13\) −2.39288 + 15.1080i −0.184068 + 1.16216i 0.706637 + 0.707576i \(0.250211\pi\)
−0.890705 + 0.454582i \(0.849789\pi\)
\(14\) −9.55523 − 6.56366i −0.682516 − 0.468833i
\(15\) 5.65667 + 9.81653i 0.377111 + 0.654436i
\(16\) −6.52218 − 14.6103i −0.407636 − 0.913144i
\(17\) −1.91459 − 12.0882i −0.112623 − 0.711072i −0.977790 − 0.209588i \(-0.932788\pi\)
0.865167 − 0.501484i \(-0.167212\pi\)
\(18\) 6.97872 + 3.32657i 0.387707 + 0.184810i
\(19\) 24.2978 − 7.89483i 1.27883 − 0.415517i 0.410663 − 0.911787i \(-0.365297\pi\)
0.868168 + 0.496270i \(0.165297\pi\)
\(20\) 18.2737 − 8.12849i 0.913684 − 0.406425i
\(21\) 13.1339i 0.625422i
\(22\) −12.6122 + 18.0259i −0.573280 + 0.819359i
\(23\) −11.4273 + 11.4273i −0.496839 + 0.496839i −0.910453 − 0.413613i \(-0.864267\pi\)
0.413613 + 0.910453i \(0.364267\pi\)
\(24\) −9.46019 + 15.4633i −0.394175 + 0.644302i
\(25\) 10.1303 + 22.8556i 0.405212 + 0.914223i
\(26\) −10.2192 − 28.8355i −0.393045 − 1.10906i
\(27\) −4.56045 − 28.7936i −0.168906 − 1.06643i
\(28\) 23.0590 + 2.41212i 0.823536 + 0.0861471i
\(29\) −16.0454 + 49.3826i −0.553289 + 1.70285i 0.147132 + 0.989117i \(0.452996\pi\)
−0.700421 + 0.713730i \(0.747004\pi\)
\(30\) −19.3134 − 11.8509i −0.643779 − 0.395029i
\(31\) −7.84453 − 10.7971i −0.253049 − 0.348292i 0.663527 − 0.748152i \(-0.269059\pi\)
−0.916576 + 0.399860i \(0.869059\pi\)
\(32\) 25.4113 + 19.4491i 0.794102 + 0.607784i
\(33\) 24.9244 + 0.214638i 0.755286 + 0.00650418i
\(34\) 14.9059 + 19.4159i 0.438410 + 0.571055i
\(35\) −3.05338 + 28.8197i −0.0872394 + 0.823421i
\(36\) −15.4404 + 0.816824i −0.428901 + 0.0226896i
\(37\) 4.00077 + 7.85195i 0.108129 + 0.212215i 0.938730 − 0.344652i \(-0.112003\pi\)
−0.830602 + 0.556867i \(0.812003\pi\)
\(38\) −35.1633 + 37.0727i −0.925351 + 0.975597i
\(39\) −20.3730 + 28.0411i −0.522385 + 0.719002i
\(40\) −24.3535 + 31.7318i −0.608837 + 0.793295i
\(41\) −75.7259 + 24.6048i −1.84697 + 0.600118i −0.849621 + 0.527395i \(0.823169\pi\)
−0.997352 + 0.0727233i \(0.976831\pi\)
\(42\) −12.5396 − 23.0814i −0.298561 − 0.549558i
\(43\) 54.5989 + 54.5989i 1.26974 + 1.26974i 0.946222 + 0.323519i \(0.104866\pi\)
0.323519 + 0.946222i \(0.395134\pi\)
\(44\) 4.95437 − 43.7202i 0.112599 − 0.993640i
\(45\) −0.995428 − 19.3019i −0.0221206 − 0.428931i
\(46\) 9.17211 − 30.9925i 0.199394 − 0.673751i
\(47\) 22.4582 − 44.0767i 0.477834 − 0.937802i −0.518727 − 0.854940i \(-0.673594\pi\)
0.996561 − 0.0828618i \(-0.0264060\pi\)
\(48\) 1.86179 − 36.2072i 0.0387872 − 0.754317i
\(49\) 9.05424 − 12.4621i 0.184780 − 0.254328i
\(50\) −39.6243 − 30.4944i −0.792487 − 0.609889i
\(51\) 8.56986 − 26.3753i 0.168036 − 0.517163i
\(52\) 45.4898 + 40.9187i 0.874804 + 0.786898i
\(53\) −21.8861 − 3.46642i −0.412945 − 0.0654041i −0.0534944 − 0.998568i \(-0.517036\pi\)
−0.359451 + 0.933164i \(0.617036\pi\)
\(54\) 35.5052 + 46.2477i 0.657504 + 0.856439i
\(55\) 54.6420 + 6.26545i 0.993490 + 0.113917i
\(56\) −42.8269 + 17.7765i −0.764765 + 0.317438i
\(57\) 57.1780 + 9.05611i 1.00312 + 0.158879i
\(58\) −18.9499 − 102.104i −0.326722 − 1.76042i
\(59\) 15.0901 − 46.4424i 0.255764 − 0.787159i −0.737915 − 0.674894i \(-0.764189\pi\)
0.993678 − 0.112265i \(-0.0358107\pi\)
\(60\) 45.2559 + 2.38726i 0.754265 + 0.0397877i
\(61\) 55.4322 − 76.2959i 0.908725 − 1.25075i −0.0588751 − 0.998265i \(-0.518751\pi\)
0.967600 − 0.252487i \(-0.0812486\pi\)
\(62\) 24.0945 + 11.4852i 0.388620 + 0.185245i
\(63\) 10.1718 − 19.9633i 0.161457 − 0.316877i
\(64\) −63.2268 − 9.91841i −0.987918 − 0.154975i
\(65\) −51.2237 + 56.7944i −0.788057 + 0.873760i
\(66\) −44.0071 + 23.4194i −0.666774 + 0.354839i
\(67\) 5.11791 + 5.11791i 0.0763867 + 0.0763867i 0.744268 − 0.667881i \(-0.232799\pi\)
−0.667881 + 0.744268i \(0.732799\pi\)
\(68\) −44.7330 − 19.8900i −0.657838 − 0.292500i
\(69\) −34.8268 + 11.3159i −0.504736 + 0.163999i
\(70\) −22.1496 − 53.5630i −0.316423 − 0.765186i
\(71\) −10.7606 + 14.8107i −0.151558 + 0.208602i −0.878044 − 0.478579i \(-0.841152\pi\)
0.726486 + 0.687181i \(0.241152\pi\)
\(72\) 26.3552 − 16.1773i 0.366044 − 0.224684i
\(73\) −46.5762 − 91.4110i −0.638030 − 1.25220i −0.952962 − 0.303090i \(-0.901982\pi\)
0.314932 − 0.949114i \(-0.398018\pi\)
\(74\) −14.5276 − 9.97927i −0.196319 − 0.134855i
\(75\) −3.05911 + 56.5659i −0.0407882 + 0.754212i
\(76\) 26.4008 − 98.7237i 0.347380 − 1.29900i
\(77\) 51.2569 + 37.9190i 0.665674 + 0.492454i
\(78\) 9.03133 − 68.7305i 0.115786 − 0.881161i
\(79\) −47.7369 − 65.7043i −0.604265 − 0.831700i 0.391825 − 0.920040i \(-0.371844\pi\)
−0.996090 + 0.0883400i \(0.971844\pi\)
\(80\) 12.5028 − 79.0170i 0.156285 − 0.987712i
\(81\) 9.66242 − 29.7379i 0.119289 − 0.367134i
\(82\) 109.589 − 115.540i 1.33645 − 1.40902i
\(83\) −9.70086 − 61.2488i −0.116878 − 0.737937i −0.974621 − 0.223860i \(-0.928134\pi\)
0.857744 − 0.514078i \(-0.171866\pi\)
\(84\) 44.0740 + 28.5912i 0.524691 + 0.340371i
\(85\) 24.9367 − 55.8832i 0.293373 − 0.657449i
\(86\) −148.080 − 43.8237i −1.72186 − 0.509578i
\(87\) −83.1957 + 83.1957i −0.956273 + 0.956273i
\(88\) 33.0350 + 81.5640i 0.375398 + 0.926864i
\(89\) 11.2250i 0.126124i 0.998010 + 0.0630619i \(0.0200865\pi\)
−0.998010 + 0.0630619i \(0.979913\pi\)
\(90\) 20.1779 + 32.9708i 0.224198 + 0.366342i
\(91\) −84.3215 + 27.3977i −0.926610 + 0.301074i
\(92\) 13.4711 + 63.2234i 0.146425 + 0.687210i
\(93\) −4.73074 − 29.8687i −0.0508682 − 0.321169i
\(94\) 2.61421 + 98.9023i 0.0278108 + 1.05215i
\(95\) 123.361 + 33.1647i 1.29853 + 0.349102i
\(96\) 31.2970 + 65.4081i 0.326010 + 0.681334i
\(97\) 7.98092 − 50.3895i 0.0822775 − 0.519480i −0.911785 − 0.410668i \(-0.865295\pi\)
0.994062 − 0.108812i \(-0.0347045\pi\)
\(98\) −4.01373 + 30.5454i −0.0409564 + 0.311688i
\(99\) −37.7185 − 19.6295i −0.380995 − 0.198278i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 220.3.w.a.63.11 yes 544
4.3 odd 2 inner 220.3.w.a.63.19 yes 544
5.2 odd 4 inner 220.3.w.a.107.24 yes 544
11.7 odd 10 inner 220.3.w.a.183.52 yes 544
20.7 even 4 inner 220.3.w.a.107.52 yes 544
44.7 even 10 inner 220.3.w.a.183.24 yes 544
55.7 even 20 inner 220.3.w.a.7.19 yes 544
220.7 odd 20 inner 220.3.w.a.7.11 ✓ 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.3.w.a.7.11 ✓ 544 220.7 odd 20 inner
220.3.w.a.7.19 yes 544 55.7 even 20 inner
220.3.w.a.63.11 yes 544 1.1 even 1 trivial
220.3.w.a.63.19 yes 544 4.3 odd 2 inner
220.3.w.a.107.24 yes 544 5.2 odd 4 inner
220.3.w.a.107.52 yes 544 20.7 even 4 inner
220.3.w.a.183.24 yes 544 44.7 even 10 inner
220.3.w.a.183.52 yes 544 11.7 odd 10 inner